Feynman integrals as Hida distributions: the case of non-perturbative potentials

Abstract

Feynman integrands are constructed as Hida distributions. For our approach we first have to construct solutions to a corresponding Schroedinger equation with time-dependent potential. This is done by a generalization of the Doss approach to time-dependent potentials. This involves an expectation w.r.t. a complex scaled Brownian motion. As examples polynomial potentials of degree $4n+2,n\in \mathbb{N},$ and singular potentials of the form $\frac{1}{|x{|}^n},n\in \mathbb{N}$ and $\frac{1}{x^n},n\in \mathbb{N},$ are worked out.

12 pages, Dedicated to Jean-Michel Bismut as a small token of appreciation